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Title Some Remarks on the Flow in Hydraulic-Machines

Author(s) Tabushi, Keizo

Editor(s)

CitationBulletin of the Naniwa University. Series A, Engineering and natural scie

nce. 1955, 3, p.9-19

Issue Date 1955-03-30

URL http://hdl.handle.net/10466/7671

Rights

9 N

Some Remarks on the Flow in IIydraulic-Machines

Keizo TABuSHI*

(Received January 19, 1955)

Abstract

With a view to obtain a rough estimation of the fiuid metion in hydraulic-machines,

equations in curved orthogonal co-ordinates (m, n, e) are given, taking the stream-line

on the rheridian section as m-axis, and some approximate relations ofthe flow are investi-

gated. Iri the first part are'described equations for the absolute flow in general (m, n,

e) co-ordinates and then in specia! (m, n, e) co-ordinates with some remarks. In the

second part,-the flow through runners is discussed. In the last part, some remarks on ,

the direct and inverse problems are given. .

1. Introduetion

For the analytical investigation of the flow in hydraulic-machines, especially in

runners, cylindrical co･ordinates or generally co-ordinates using parallel conical surfaces

are preferable. They are, however, inconvenient for the rough presumption of the fiow

and the quick grasp of the results obtained because the equations become very com-

plicated for the three-dimensional flow.

On the other hand, the curved orthogonal co-ordinates, taking a stream-line on the

meridian section as an axis, appear better suited for the approximate estimation of the

flow in hydraulic-machines and the design of three-dimensional runners, so investigations

are made on some properties of the flow by these co-ordinates. ' ' ' 2. Equations for the absolute fio.w in curved orthegonal codinates

' As showfi in Fig. 1, the zaxis Qf symmetry is taken as

2-axis, two curves on the me- -r

ridian plane intersecting at

right angles at any pointPare P

taken as m- and n-axes and the

circumferential direction th-

rough P is taken as e-axis,

thus curved orthogonal co-

ordinates (m, n, e) with P as

origin is determined. The cylindrical

When the external mass force has

equation holds

ct

B

A

. d

co-ordinates

the

.* Department of Mechanical Engineering, College

za ,n ' d" 1af Bx m R'"..,A' e .B.. 'c;'7 iiiile %.''E"c

C ix' A ,.t'..lll)! 211I' ;¥;l `eN ,g,

li:, B' IZ: .R 'B,

n ' Fig. 1. of P is expressed by (2, ag e).

potential 2, the following wellknown vector

'

of Engineering.

10 . Keizo TABusHi :-Ct--cxg :-grad(g+g+-S2 -vrote ････････････(i) )

where c denotes the absolute velocity of the flow, P the pressure, p the density, v the

kinematic viscosity, e the vorticity of the absolute flow. If the unit vectors in m, n and

e directions are expressed by ii, i2, and i3 respectively, then

} = rot b = ii6m+i2en+i3ee

e.=-i,{8.(rce)-OoCe"}, e.==-IL{OoCom-O(ar;ie)}, 6e=g;ii"-aoC.m ････････････(2)

Therefore, the equations of motion in m, n, e directions are

' atCm/ +c. OaC.m+ce 9gr-c. 3£i" -ce to.te-tt i2 sina : - o-O.(-il- +g+S2 )+yKin ･ "'''''''(3)

%C-t"-+c.3iZ"+ce %"e-c.{lit);nm-ce 31/ee-fr'}2 cosa= - -oOsii(e+2+!li2-)+vKh ･ ･････････(4)

ao-C-te + c.aO-jl'i +c. gl/ee - c.O-rSz - c. 9-S-"o +g!t;lti- sin cx + CerC" cos a =: - rge(-{} + 2 + C-22 ) +yKb .

-･-･-･･-(5)

.where a is the angle between 2- and m-axes and

sina=toi, cosa= 3: ････････････(6)

K;.-72c.-3frCiy-S'","gllmi-90;`V3i£h'-,9.(31'£.")-,g,(3-ji';)-Si",a9-,Cz--!ig'-O-,Sij/I"-.(7)

Kh=72c.-Oo.2C,"-COS,aaoC."-Si".aOoC.m-tjl.r(OoC.m)-roOe(31/4e)'CO;a9oCbe-C-,eOC,oO-S-o--gl.(s)

Kb = 72ce- 9,-o2Cie2 - -li- bOiiim(t-oCege) +Si; a %z +op iSml (Sinr a) + ce zS}i(COrS at)

-i,-8.(lld'o")+CO;"9S3 ･-･-･-･-(9)

72=(oOmt2+oOn22+r20o2e2.+Sinrao-Om+CO;azlltT) ････････････(10)

(3), (4), (5) are also expressed as follows;-

' OoCmt +c.OoCfi+cn OoCnin+ce %z-V'2 sina = -oeOm(-{lm +2) +vK;n ............(11)

{lit;t" + cmZiil'i + cn tblt; + ce 9g"e - 3I2 cos cv = - o-On ({l- +g) + vKh ････････････(12)

) tbltO+c.gty;z+c. 3ti}'+ce -Orlg/t+CeCrm sina+CO "cosa= -roOo(-;-+g +vKb ･'''････''''(13)

The left hand sides of (11), (12), (13) can be expressed as Dc./Dt, Dc./Dt, Dc/Dt

respectively. The condition for continuity is

divi=O , ････････-･･･(14)

' or Ou(a'-liim)-+51./(rcn)+tha'e=o ''''''''''''(is)

Some Remarles on the Elow in HZ!,draulic-Mdchines 11

or

g,C.in+ZS"+%ee+C-.M sina+ fil'L cosa ==O ････････････ a6)

Equations (11), (12) are obtained also from momentum equations in m, n directions

using (16) and the last term on the left hand side of each is introduced because the

the sides C and C' or e surfaces in Fig. 1 are not parallel each other. Equation (13) is

obtained from the equilibrium of the moment of momentum about 2 axis.

When the flow is turbulent, we denote the component velocity cke (a==m, or n, 0),

as ca=cao+cd', where ckeo=the time mean value of ca, czx'=the variable part df cd.

Equations (15) and (16) hold for both ca, and c.'. Time mean Values are.obtained by

(11), (12), (13) with some modifications, namely, on the left hand side the first term

dissappears and the rest does not change its form, but become the sum of two groups,

one is expressed witfi cao for ca, the other cd' for cd of original equations, and on the

right hand side each term takes the time mean value keeping the original form.

Similarly we can obtain 'from other equations the expressions for time mean values.

cin-Stream-Line taleen as m-Axis

For the study of the flow in hydraulic-machines, it is convenient to take the

absolute stream-line on the meridian surface (we will call it cin-stream-line for the pre-

sent) as m-axis. Then

c.-o, 3I"fsc.3//i, Z-;"fec.g:, 9Szfscin9,", ････････････(i7)

The equations of motion expressed in terms of the normal stress a and the tangential

stress T are obtained from the equilibrium of forces on a volume element as shown in

DDC7=-3/i/l+p-i,{{2-l/o'.Vm)+O(roT."m)+OoTeem}--pi-[o.tb"li+aeSi".`V+T..Oo-".] ･････････(is)

Ditn = -lll/l+i{O(o'a.n)+O(or.Tmn)+OoTeen}--}-[aeCO.Sa-o.gtin-T..g--".] ････････････(ig)

PD-Ctp = -9a9o+iYr{oOoae+O(roime)+P(.rorn"e)}+-l;l[Te.Si"ra- Te. ,cosr `u] ･･r････････(2o)

Terms in the last brackets on the right hand side are introduced from the condition

that any pair of surfaces A and A' (or m-surfaces), B and B' (or n-surfaces) or C and

C' in Fig. 1 are not parallel, and terms containing Oa/On, Oa/Om within them are the

effect of A, A' and B, B' respectively. .

In most cases, however, the change of a in e and n directions is not large and we

can use equations (2)･w(14), putting the relation (17) in them, as approximate expres-

sionsi in such cases. ' As n-surfaces are composed of cm-stream-lines, there is some movement of the fluid

through them when Ocr/OO is not zero, but the magnitude of this fiow is generally

negligible compared with those through A, A' or C, C' in Fig, 1 so the condition of con-

12 Keize TABusHitinuity is expressed approximately as follows;

o-O.(':1'Cm)+,oOo(,:1'ce) ==O ''''''''''''(21)

where 1' denotes the thickness of the volume element inndirection. From (21)astream-

function ip.' can be introduced. Thus

i'c.= 9oOen', f'6,= -gg.b' ････････････(22)

3. Equations for the fiow through runners

General Cage

Let e = the direction of the rotation of runner,

to = the angular velecity of the runner, u=rw the velocity of rotation, tv=the

relative velocity. ･When tu is constant and the relative fiow is stationary, the variation of b in time inter-

val dt at any space fixed point is approximately equal to the difference of c between

two points which are udt=-rde apart in e direction at any instant in that time interval,

thus

--,a･t-TC-e 'udt = -to g// dt .

'When the relative fiow is not stationary, a term (Oth/Ot)dt is added to the above quantity.

Theaefore in general

tbl'=-tuoO-eCT+{li-:'t and -bxF=-abxe-'u-×4,

- ti ×e = - ti ×g-i,a(,'o$) + i,tu ll+e = -grad (uce) +tu 8// ･

From above conditions the equation Qf motion (1) is transformed as follows;

toWt -di ×e = -gr.ad(g+2+g2- uce) -v rot } ･･･-････････(23)

If the external mass force is the gravity force .

a=gh ･････････-･(24)

where h is the height in the direction opposite to g. From the volocity triangle

w2 u2 C2 ･･･････-････(25) 2 -UCe = -2---2L

arid putting

ng =r, i= e +h+ tWi-t",2 -･----･(26)

Equation (23) becomes

tbl'- ab xg = - grad (gl)-v rot5 ･ 'm m""' (27)

Some Remarfes on the ')Fleotv in fl3,draulic-imchines 13

Also foliowing relgtions exist

S == rot th+2di= t+2w-' ･･-･････････(28)

rot}=rott+A, A =iiAm+i,An+i3Ae "''"'"'''(29) Am =2caO(/ronoa)', A. =2tuO(CrOoSaa), Ae == -2w(O /j;lla+aCoOnS ev).

Let

2 P- P2" --- P ････････････(30)

then, from (23), (25), (28), (29), (30) we have

21ti-cb×ij I -grad(iPff+g+!ll2L -yrott-(2di×to+yA-) ････････････(31) )

If we neglect the term (2di×nd+yA- ), Equation (31) has a similar form as Equation (1),

or (31) can be obtained putting nd and P in Place of 'c and P in (1), When the visco-

sity or the change of a is negligible, (2di×nd+yA)fu2tu-×to represents the Coriolis ac-

celeration, and we can say that if (P-pu2/2) of the relative fiow in a rotating runner is

considered just as P of the fiow in the same runner at rest, the difference of the two

fiows is caused mainly by the Coriolis force.

wm-Stream-Line taleen as m-Axis

For the computation of the flow in a runner, it is convinient to take co-ordinate

axes rotating with the runner. In this case, the relative stream-line on the meridian

surface or zvm-streamLline is taken as m-axis and the direction of rotation as e. Equa-

tions of motion in m, n, e directions are derived from Equation (31) etc. Neglecting the

fiow through n surface of a volume elernent similar as in Fig. 1, the condition of conti-

o-O.(,:1'wm) + ,oOe(2:1'we) = O ''''''''''"(32)

'and this is satisfied by a stream functiQn ipn.

]'wm=9aipe", ]'we=-ao-¢mn-- ･･･････-････(33)

When n-surfaces are constructed from relative

stream-lines or relative paths passing through er'a. wm-tshter:al:}tieiinfleo;}n anY'defi%teeseMg:irdfia'acnesSgg'". e nj' -.;li at.- iSIIIi E;:r

dle) ig l N ,,,

tion, throughcemes identically equal to zero and intersection

curves of these surfaces and other meridian

sections do not generally coincide with wm-

stream-lines on these sections if Oa/rOe=t:O.

Such surfaces are importent when I or H of the

fiow varies in the n direction.

In Fig. 2, Pb"JFInt" represents a relative stream

.oP C-ll J)o

... -LkP･

er -." x XN R

n

Fig. 2

ag

ty

14 Keizo CABusHr line and the surface (PP;."Ph"Pb") is composed of such lines, PP;n, PbiP'are zvm-

stream-lines and surface (PJe,s P;t' Pe) is composed of wm-stream-lines.

.Then

･ ::Iilt == hadOff- ww.e. ,. From the geometrical relation of the projected figure of the surface (PP."jP?,"Pb")

(which is assumed ta be an element of a plane surface) on the meridian section, we

have

PePe"s fPRn" sin 6fsdin tan 6f frde tane

:. tane=tan6lldlil =-tan6wW-eM ･･''"''''''(34)

313 ugi/Y - :Ilam tan6sin (a-･6) ･･-･･:･･････(3s)

If tvm/zve, a, Oa/OO and the initial value of 6 are known, the values of e, 6 can be deter-

mined from (34), (35). Above relations are also obtained graphically.

Flbw of ldeal Fluid

When v=O and Ow/Ot=O we have from Equation (27)

nd×6 =grad (gl) ･････････-･･(36)

Equation (36) means that both thand 6 are perpendicular to grad (gl) or l is constant

along the relative stream-line.

' If wm=l=O, we have from the n component of vectors of Equation (36)

OotV.m==wmll/in+we.Ooae+.-1.o-a.(gl)--l)-:I:O(a'T.O) 'm"'""'(37)

and this is a condition for the velocity distribution on the meridian section.

If gl is constant on the n-surface and this surface is composed of relative stream

lines, then grad (gl) becomes equal to i30(gl)10n and from (36) 6n=O, and this last

condition (which is the same with the potential flow) determines the velocity distribution

on the n-surface.

Potential Flow' ,

In'many cases the absolute flow through runners is considered to have started from

an irrotational condition, so it can be treated as a potential flow if no free vortices are

shed from wall surfaces. In this case 6= O, Z =-2di ; and if Odi/Ot=O, we have from (36)

grad (gl)= O, gl=const ･-･･････････(38)

If there exists any exchange of energy between the runner and the fluid, the total head

'Hof the flQw must be changed where ' H=p/r+h+c2/(2g) ･･････-･････(39) From (38), (26), (25),

.

thme Remarfes on the rvoto in Hb,draulic-Atftvchines 15

grad (gH)=grad (uce) ････････････(40)

The same result can be obtained from (1) as follows;

Ob Ond Oj b7/ == 57t -tu brt

and from l=O, tuab/OO=grad (ucJ), moreover Ow/Ot =O, y=O, so Equation (1) takes the ' 'form (40).

In ordinary runners grad(gH)ikO, grad(ucE)=VO, O(rce)/am=l=O, therefore from

en= O, Otvm/OO =kO.

We can say from this last relation together with boundary conditions on blade sur-

faces that in runners P also varies in 0 direction and by this rotating pressure field, the

energy is transmitted from the runner to the flow or vice versa, and in this state the

rotating blade or the rotating vortex surface gives the change of (rce) etc, and becomes

the discontinuous surface of the pressure and the velocity field. The velocity distribu-

tion on the n-surface can be obtained from boundary conditions and 6n=O, the latter is

also expressed, assuming Ol'/rOestsO, Ol'/Omftrj04/an as follows ,

aorm2.di,n+,O,2oipon,+g$n(Sin.'a-tb;'i)fe2ojsinat. ････････････(4i)

If Oa/rOe =O, the above relation holds for the whole domain of n-surface, but in

other cases it can be applied to a narrow annular area on the n-surface through any

(2, r) position. The velocity distribution on 0- and m-surfaces can be obtained from the

ttboundary conditions and the following relations

0-surface ;

6e == tv. 31g.2 - aoW.m == o ･･･････`････(42)

' t. m-surface :

' ･1･･g. = -li-{O (orie) -- w. ao--ae} =o ････････････(43)

If there is no exchange of energy between the runner and the fiuid, H and (rce)

become constant, w. and we do not vary in e-direction, so the fiow is axisymmetric.

The. wm-stream-line in this case can be determined graphically or experimentally by

wellkhown methods, and this stream-line or its inclination a can be adopted as the first

approximatlon ln varlous.cases.

Comparing this fiow and the potential (absolute) fiow with energy exchange, Equa-

tion (42) remains the same but the stream-line on e-surface is not the same, for the

distance of streamlines and the value of wm is not the same.

' Stream-lines of both flows, however, become the same if in the potential flow with

energy exchange (rce)=constant in the n direction on every e-surface, because in this

case Oa/OO becomes zero from (43). On the contrary, the stream-line on the 6-surfece

may separate from the wall if (ree) distribution is not appropriate,

the original co-ordinate axes and

only take the change of a and wn

(if it reaches to a considerable

magnitude) into account.

When the flow has variable gl at

the friction of upstream canal etc., the

so we can proceed the same way as

mation of surfaces with constant gl'

needed. Now we make remarks on

Assuming .Oa/Oe vO, the flow on

plane surfaces such as (R, e) plane. or

e) co-ordinates and (R, e) polar

following relations;

dm clR rdO P Rde -

e

16 Keizo TABusHr 4. Remarks on the direct and inverse Problems of three dimentional runners

The direct problem or the problem for the estimation of the flow character of a given

runner and the inverse or design problem are not simple for three dimensiona! runners.

There are some ingenious methods') for solving these pr6blems adopting cylindrical co-

ordinates, but in rnany cases they are somewhat complicated and inconvenient for a rough

but quick estimation. For the first rough estimation of above problems, the (m, n, e) co-

ordinates seem to be better suited. A method by these co-ordinates is as follows;

we take firs.t the (m, n, e) co-ordinates of an axisymmetric potential flow, then assuming

the absolute flow has the velocity potential, require the flow (zvm, we or ¢n) on several

n-surfaces from conditions of boundaries, of the irrotation and the continuity. In this

process the transformation of n-surfaces upon plane surfaces may be needed. For the

purpose of a very rough estimation, the flow on the transformed plane surface may be

approximated as a two-dimensional fiow. The next step is to check the velocity distribu-

tion on some e-surfaces and alsothe quantity of flow asawhole (a) e-S"r3F"Ce (b) (R,e)-p{ane

and the condition (46), etc., as z

described below. For many en- r 7ngineering problems only above 2steps are suthcient and the (m, n, .P 3 ･1e) co-ordinates are very effective -5urf

for them. When more precise nresults are needed, the flow on n- nTS"if

'surfaces are again treated with (c) (S,¢)-planeimproved wm,we,]' , a, but in Fig. 3this case it is preferable to retain

P2

P

'R,

@

¢ t-st

the inlet of the runner in the n direction due to

absolute fiow on the n-surface remains irrotational,

described above and the computation of the defor-

is only necessary when more detai!ed results are i some of the above subjects.

the n-surface can be transformed conformally on

(S, ¢) plane as shown in Fig. 3. Between (m, n,

co-ordinates or (S,¢) rectangular co-ordinates exist

' ' ' dS lv. -- tvR - ws' -do' ib5 -'riiJle'ilJJdi' '''"''"'''(44)

Some Remarks on the rvow in lfydraulic-Atlachines 17

putting e/e =x, ¢/@=x', ¢/e =x"=xxl we have i

£til = exp {x Sr {!IIZ} , S- s, = xt' Sr dll! , fir, = exp {S -i,Si} . ･････････ (4s)

Sr means the integration along wm-stream-iine in the domain considered. Then Equation

(41) is transformed inte following forms ;

' 3'-k4+R-'gk'(i-f3-".)+.022oipe2=2to'･ ''''''''''''(46)

gk-g-.-r,,3/I3Ls,g+g,2$-2,,n, ････････････(47)

' where ip=ipn/1'me, ]'me == Si7'dw/Si din

' of =w( xi5 )2(i.mi'e) sina

of' : tu(x-rt/)27.m7'esina = tu'R2(lfi,J,)2 =tot(Sl,)2

'S? means the integration alorig the stream-line covering th6 total part of the flow con-

sidered. Velocity components and relations between them are (if S, ¢ are considered

dimensionless) ;

tv.=(i'itl!e)rOoipe, tve=--(j'Mi.e)gilh/, tvR= Soipo, we=-lllRlt, tvs= 31$-},

wdi=-･2g--}, ..m.･=jM,.e"-.R-.W-g. ---･--(4s)

ggLm ., jmie x-'Le .. eqg ............(4g)

ws 7r wipwhere e means unit length.

When (acr/On)=O and tu'=const, Equation (46) represents the two dimensional flow

in a runner rotating with angular velocity of and whose absolute velocity has a velocity

potential. Similariy, if (aa/On)=O and of'= const, Equation (47) represents the general

relation of the stream function of two dimensional fiow whose absolute velocity has a

velocity potential. But, as tu" is a function of of, x' and R, the fiow in (R, e)-plane

and its transformed flow in (S, ¢)-plane can not become potential flows simultaneously.

Moreover, in the present case, the transformed runner vanes form a straight row on the

(S, ¢)-plane and the potential fiow through these vanes satisfies the condition that Oa/

On=O and tu"=O or a=O in Equation (47). a=O means an axial flow in (m, n,e) co-

ordinates. The magnitude of x, x' or x" can be chosen at any convenient value.

The condition of similarity of the velocity taingle on both (m, n, 0)-and (R, e)-surfaces

is x= sinev. The condition that the ratio of any corresponding length at sections 1 and

2 remains the same for both (m, n, e)- and (R, @)-surfaces is x ={log.(r2/ri)}/Si(cim/r).

The transformation on (R, @) plane serves for the rough estimation of the fiew and

the boundary stream-lines if the fiow is treated as two dimensional, ,and the transforma-

tion on (S, ¢) plane may be useful fQr the rough numerical calculation of ¢.

)18 Keizo TABusHi On the blade surface, following conditions hold. If the blade itse]f is considered .ttOheb.e reCfO.MrrPiO.Sgedt.OkVgO. rEi,C:glllil4ilt,h..gCOrMelPaOtnioenn,tSeE;,L.C", Ce 'n m, n, e directions respectively,

Cm = len(tVedm-Web)/v/le.2+ ke2 == (WedmZVeb) sin a',

Cn = {lee(ZVmd- Wmb) h lem(Wed-- ZVeb)}/i/'lee2 + k.2 == (WebmWed) COs B' mu (ZVmb- ZV.d) sin Bl

Ce = len(ZVmbhZVmd)//k.2+le.2 = (Wmb-' ZVmd) COS r'

where fem,kn,lee, are direction cosines of the normal to the surface, al, B', r' are the

angles between n,e,m axes and the section curves of the blade on m-, n-,e-surfaces

respectively, suMxes b, d mean the quantities at the back and front surfaces of the blade.

A 't'-`- Let in Fig. 4 io ao be the entrance profile of the blade, a' i' be a part of the blade

.t--x -Asection on a e-surface, io' i' and ao' a' represent blade sections on two n-surfaces,

then

Si･:, Cndsn-' S::, Cndsn =S,".,' c dsj ････-････-･･(50)

where dsn and clse are elementary lengths of blade sections z

on n- and O-surfaces respectively. There exist also condi- erk 'M

tions similar to those usually considered in two dimensional Z v

cases such as; relations between Pb, Pd, tob, z-vd and gl: the L v or

direction of te and the value of ¢ on blade surfaces;4 and g 9nd at blade tips, etc, The conditions for the fiow asawhole / Ci{ e

are ; the total quantity of flow at'any section obtained from

the velocity distribution must be constant; the torque ob- e d 'M ctained from the pressure distribution on blade surfaces and

that from the velocity distribution around the runner must

coincide ; the relation between the circulation around a blade

section on the n-surface and velocity distribution on the

same surface around the runner.

Direct and inverse problems are often solved by trans-

forming differential equations into difference equations and applying

or the relaxation method3)., For this purpose dimensionless form of

Runners can also be designed by assuming the mean value of

surface, and this value is given to the middle portion between two vanes

distribution is obtained by extending the domain of known quantities')`)

of the flow. In such cases the following relations may also be used

･ e. ==o, O(or.Cb) .. OoWom

'

the condition of continuity is from (16), (17),

' a(orece) = -- r2 {Oowmm tw.(si"r "t + 23 I)}

p' 6 A o, Td di b d

Bn Fig. 4

the matrix method

(47) may be used.

O(rce)/Om on any n-

and the velocity

uslng equatlons

..･･･････-･･(51)

'

-･････-･････(52)

r

Some Remarks on the Elow in H3,dra"lic･Mbuchines 19

.If the wm distribution of the axisymmetric potential fiow is taken as the mean value of

tvm, the left hand side of (51) and the right hand side of (52) are known for the

middle portion between two vanes and the variation of zvm and (rco) in e direction can

be estimated at this portion. From (51), (52) we have a differential equation for wm,

' ･ Oo2.W,m +9i6Wom, +06W.m (A,+B)+w.(AtB,+ct) = o ････････････ (s3)

where A' =Sinra +23t;I, B' =2Sirna, c' = OoAi.

The separation of fiow from wall surfaces may be caused by the inadequate form of

walls and the effect of the viscosity. The former can be expressed as the effect of an

inadequate ditribution of (rc) and in some cases may be estimated from the flow of

ideal fiuid, while the latter is computed from boundary layers. A rough idea of these

boundary layers is obtained from boundary layers on axisymmetric walls and rotating

canals for which Equation (31) may be of use.

References

1) C. H. Wu, Trans. ASME., 74, 1363 (!952).

2) C. H. Wu, NACA TECH. NOTE 2214 (1950).3) R. V. Southwell, "Relaxation Methods in Theoretical Physics", Clarendon Press. (Oxford).

4) KC. Ho, R. J. Moon, j. of Appl. Phys. 24, 1186 (1953).